AP Calculus BC Flashcards: The $n^{th}$ Term Test for Divergence
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What is the first step in applying the $n^{th}$ Term Test for Divergence to a series $\sum a_n$?
The first step is to find the limit of the $n^{th}$ term as $n$ approaches infinity, i.e., calculate $\lim_{n \to \infty} a_n$.
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What is the first step in applying the $n^{th}$ Term Test for Divergence to a series $\sum a_n$?
The first step is to find the limit of the $n^{th}$ term as $n$ approaches infinity, i.e., calculate $\lim_{n \to \infty} a_n$.
For which AP Calculus course is the $n^{th}$ Term Test specifically listed as a topic?
The $n^{th}$ Term Test is listed as a topic for AP Calculus BC only.
What is the $n^{th}$ Term Test?
The $n^{th}$ Term Test is a method used to determine if an infinite series diverges.
What is the primary goal when applying a test like the $n^{th}$ Term Test to a series?
The primary goal is to determine whether the given infinite series converges or diverges.
What does the $n^{th}$ Term Test conclude if the limit of the $n^{th}$ term of a series IS zero?
The test is inconclusive. The series may either converge or diverge, and another test must be used.
Using the $n^{th}$ Term Test, what can you conclude about the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$?
Since the limit of the $n^{th}$ term is 0, the $n^{th}$ Term Test is inconclusive for this series.
Using the $n^{th}$ Term Test, what can you conclude about the series $\sum_{n=1}^{\infty} \frac{n}{2n+1}$?
Since the limit of the $n^{th}$ term is 1/2 (which is not zero), the series diverges by the $n^{th}$ Term Test.
What conclusion can be drawn if the limit of the $n^{th}$ term of a series is NOT zero?
If the limit of the $n^{th}$ term is not zero, the series diverges according to the $n^{th}$ Term Test.
Why is the $n^{th}$ Term Test often the first test one should apply to a series?
It is often a quick and easy way to determine if a series diverges before trying more complex convergence tests.
Can the $n^{th}$ Term Test ever be used to prove that a series converges?
No, the $n^{th}$ Term Test is only a test for divergence; it can never be used to prove convergence.